Let us note $\overline{\mathbb{C}}$ the Alexandroff compactification of $\mathbb{C}$ (i.e. the Riemann Sphere). I can prove that $$H^1(\overline{\mathbb{C}},\Omega_{\overline{\mathbb{C}}}) \simeq \mathbb{C},$$ where $\Omega_\overline{\mathbb{C}}$ is the sheaf of holomorphic one-forms. However, I would like to prove that the integration on $\overline{\mathbb{C}}$ : $$\int : H^1(\overline{\mathbb{C}},\Omega_{\overline{\mathbb{C}}}) \to \mathbb{C}$$ is an isomorphism. I tried some Mayer-Vietoris sequences but can't get the conclusion.
Any help will be appreciated.
Edit I think I can prove surjectivity. Consider $\omega_0 = \frac{1}{2i\pi}\frac{dz}{z}.$ Then $\int [\omega_0] = 1$ and hence for all $a \in \mathbb{C}$ $$\int a [\omega_0] =a.$$ The injectivity remains a problem, how can I prove that $\int \omega =0$ implies that $\omega = \partial \omega'$ for a $\omega'$ ?